


In Which Crowley and Aziraphale Debate Non-Euclidean Geometry

by DoctorTrekLock



Series: AU-gust 2020 [21]
Category: Good Omens - Neil Gaiman & Terry Pratchett
Genre: Gen, Human AU, Mathematics, Modern AU, Non-Euclidean Geometry, Professor AU, the Parallel Postulate
Language: English
Status: Completed
Published: 2020-08-22
Updated: 2020-08-22
Packaged: 2021-03-07 01:48:11
Rating: General Audiences
Warnings: No Archive Warnings Apply
Chapters: 1
Words: 960
Publisher: archiveofourown.org
Story URL: https://archiveofourown.org/works/26038942
Author URL: https://archiveofourown.org/users/DoctorTrekLock/pseuds/DoctorTrekLock
Summary: “The Parallel Postulate,” he announced when he was finished, capping the marker and turning on his heel with a flourish. “Euclid’s Fifth Axiom. Who can define it for me?”It had been in the reading, and Aziraphale was heartened to see a dozen faces who looked confident in their answer. Only Anathema, seated in the second row, raised her hand, however.“Yes?”“The Parallel Postulate states that for every lineland pointpnot lying on linel, there exists one and only one line that both contains pointpand does not intersect linel.”“Correct. Now--” Aziraphale leaned forward “--what does thatmean?”
Relationships: Aziraphale & Crowley (Good Omens)
Series: AU-gust 2020 [21]
Series URL: https://archiveofourown.org/series/1870924
Comments: 2
Kudos: 24





	In Which Crowley and Aziraphale Debate Non-Euclidean Geometry

**Author's Note:**

> Originally posted August 21, 2020 on [Tumblr](https://doctortreklock.tumblr.com/post/627119456654589952/au-gust-21-professional-rivals-au)

Aziraphale hummed under his breath as he erased the whiteboard in the front of his classroom. He had to admit a bit of melancholy at the loss of the chalkboards that had once been so ubiquitous. To his mind, nothing quite said _learning_ like the smell of chalk dust in the air. However, needs must.

And, he had to admit to himself as his hands hovered over the wide array of markers at his disposal, chalk never came in quite such a variety of brilliant colors.

He plucked a blue one off the ledge and began to refill the board with his neat lettering, perfectly practiced to ensure maximum legibility.

“The Parallel Postulate,” he announced when he was finished, capping the marker and turning on his heel with a flourish. “Euclid’s Fifth Axiom. Who can define it for me?”

It had been in the reading, and Aziraphale was heartened to see a dozen faces who looked confident in their answer. Only Anathema, seated in the second row, raised her hand, however.

“Yes?”

“The Parallel Postulate states that for every line _l_ and point _p_ not lying on line _l_ , there exists one and only one line that both contains point _p_ and does not intersect line _l_.”

“Correct. Now--” Aziraphale leaned forward “--what does that _mean_?”

There was a pause. Then, from the side of the room, one of the sophomores put his hand in the air. “Parallel lines are unique?” Wensleydale guessed.

Aziraphale nodded slowly. “That is a more or less logically equivalent statement,” he allowed. “But what does it mean that this is a _postulate_ , and not a theorem?”

The pause was longer now as his students mulled over the question. Then Adam Young stuck his hand up from the back of the room. “It hasn’t been proven.”

Aziraphale smiled and tapped his nose in delight. “That’s it exactly! In fact, mathematicians like Gauss have proven it to be _unprovable_!”

He beamed at the class, but received only a few looks of comprehension, and the rest of the students looked lost.

“The important point here,” he told them, trying not to look too discouraged, “is that the Parallel Postulate cannot be proven based on our current set of theorems. Instead, we must set it as an axiom.”

 _He_ might get a thrill from understanding the complicated way postulates and theorems fit together within the rules of mathematical logic, but for many of his students, the goal was a general understanding of the course material and a passing grade.

“As you may have noticed, the model of geometry we primarily work in - the Cartesian plane - holds that the Parallel Postulate is true. Parallel lines are unique, as Wensleydale pointed out.”

Aziraphale was now getting enough nods and looks of comprehension that he felt comfortable with where he was leaving it.

“Since the span of human observation has agreed that the world is Euclidean, for the purposes of this class we will be assuming--”

“Oh, come off it!”

Aziraphale broke off mid-sentence, turning his attention sharply to the source of the disruption.

A dark figure lurked on the threshold of his classroom, where he’d left the door ajar to help with air circulation. It was one of the newer teachers, an adjunct filling while Dr. Beelz was on sabbatical.

Aziraphale straightened up as much as he could. “Professor Crowley,” he said coldly. “I do not believe I recall inviting you to observe my class.”

The students watched the pair quietly, eyeing Crowley in curiosity. Aziraphale knew most of the students wouldn’t have had a chance yet to meet him, but he didn’t feel charitable enough at the moment to make the introduction.

Crowley ignored Aziraphale’s remark. “Don’t listen to him,” Crowley advised his class. Aziraphale bristled. “Just because the world _looks_ Euclidean doesn’t mean it _is_.” Crowley looked up and met Aziraphale’s eyes. “The world looks like a lot of things at first glance that don’t turn out to be remotely true.”

Aziraphale narrowed his eyes. “Mathematical models are meant to allow us to describe the space around us, particularly in geometry. There is no point to using overly-complicated hyperbolic models when Euclidean geometry is clear, simple, and reflective of the universe around us.”

Crowley snorted - actually _snorted_ \- at him. “That’s such a narrow way to look at mathematics,” he said. Then he tipped his head and held Aziraphale’s eyes, his gaze almost hypnotic.

“Haven’t you ever looked at a non-Euclidean model?” he asked, his voice low, the moment surprisingly...intimate for one happening across his classroom with an audience of two dozen math majors.

“Sat down with a ruler and compass and started doodling Klein disks just so you could calculate the angles?” His eyes were very bright, Aziraphale noticed, almost gold, and vibrant in his animation.

“It’s just so much more _elegant_ , don’t you think? Do you ever just stop and look at a Poincaré disk and marvel at the wonder of the universe?”

He hadn’t, Aziraphale thought. He’d always found non-Euclidean geometry to be too fussy, preferring infinite lines in infinite planes to working within a bounded space with complicated rules and equations.

But with Dr. Crowley looking at him like that, Aziraphale had to admit to himself that there was some poetry in the idea of an infinite world trapped in a finite shape.

Before he could open his mouth to try and respond, or to try and futilely remind his students of the problem set they had due next week, he was interrupted by the shuffle of feet and murmur of conversation in the hallway.

Aziraphale glanced at the clock. Class was over. His students were already packing up and heading out of the room.

When he looked back at the door, Crowley was gone.


End file.
